Optimal. Leaf size=157 \[ \frac{a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}-\frac{a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac{\sqrt{3} a^{4/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )+\frac{3 a p x}{4 b}-\frac{3 p x^4}{16} \]
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Rubi [A] time = 0.114458, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2455, 302, 200, 31, 634, 617, 204, 628} \[ \frac{a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}-\frac{a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac{\sqrt{3} a^{4/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )+\frac{3 a p x}{4 b}-\frac{3 p x^4}{16} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 302
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int x^3 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{4} (3 b p) \int \frac{x^6}{a+b x^3} \, dx\\ &=\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{4} (3 b p) \int \left (-\frac{a}{b^2}+\frac{x^3}{b}+\frac{a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{3 a p x}{4 b}-\frac{3 p x^4}{16}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (3 a^2 p\right ) \int \frac{1}{a+b x^3} \, dx}{4 b}\\ &=\frac{3 a p x}{4 b}-\frac{3 p x^4}{16}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (a^{4/3} p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{4 b}-\frac{\left (a^{4/3} p\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 b}\\ &=\frac{3 a p x}{4 b}-\frac{3 p x^4}{16}-\frac{a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )+\frac{\left (a^{4/3} p\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 b^{4/3}}-\frac{\left (3 a^{5/3} p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 b}\\ &=\frac{3 a p x}{4 b}-\frac{3 p x^4}{16}-\frac{a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac{a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (3 a^{4/3} p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{4 b^{4/3}}\\ &=\frac{3 a p x}{4 b}-\frac{3 p x^4}{16}+\frac{\sqrt{3} a^{4/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{4 b^{4/3}}-\frac{a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 b^{4/3}}+\frac{a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 b^{4/3}}+\frac{1}{4} x^4 \log \left (c \left (a+b x^3\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0495574, size = 147, normalized size = 0.94 \[ \frac{2 a^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-4 a^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+4 \sqrt{3} a^{4/3} p \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+4 b^{4/3} x^4 \log \left (c \left (a+b x^3\right )^p\right )+12 a \sqrt [3]{b} p x-3 b^{4/3} p x^4}{16 b^{4/3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.513, size = 194, normalized size = 1.2 \begin{align*}{\frac{{x}^{4}\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{4}}-{\frac{i}{8}}\pi \,{x}^{4}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,{x}^{4} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{8}}\pi \,{x}^{4}{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}-{\frac{i}{8}}\pi \,{x}^{4} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ){x}^{4}}{4}}-{\frac{3\,p{x}^{4}}{16}}-{\frac{{a}^{2}p}{4\,{b}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{2}}}}+{\frac{3\,apx}{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04393, size = 356, normalized size = 2.27 \begin{align*} \frac{4 \, b p x^{4} \log \left (b x^{3} + a\right ) - 3 \, b p x^{4} + 4 \, b x^{4} \log \left (c\right ) + 4 \, \sqrt{3} a p \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) - 2 \, a p \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 4 \, a p \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 12 \, a p x}{16 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28597, size = 216, normalized size = 1.38 \begin{align*} \frac{1}{8} \, a^{2} b^{3} p{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a b^{4}} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b^{5}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b^{5}}\right )} + \frac{1}{4} \, p x^{4} \log \left (b x^{3} + a\right ) - \frac{1}{16} \,{\left (3 \, p - 4 \, \log \left (c\right )\right )} x^{4} + \frac{3 \, a p x}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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